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The normal from origin$\left(O\right)$ to a line meets it at the point $P.$

Let $OP=2$ and $\angle POX=\alpha .$ The line meets the co-ordinate axes at $A$ and $B$ respectively, then the locus of mid point of $AB$ is

Two newspapers $A$ and $B$ are published in a city. It is known that $30\%$ of the city population read $A$ and $25\%$ read $B$, while $10\%$ read both $A$ and $B$. Further, $20\%$ of those who read $A$ but not $B$ look into sports news and $40\%$ of those who read $B$ but not $A$ also look into sports news, while $50\%$ of those who read both $A$ and $B$ look into sports news. Then the percentage of the population who look into sport news, is

If a complex number $z$ satisfies $|2z+10+10i|<5\sqrt{3}-5$, then the least principal argument of $z$ is

Evaluate $\int \frac{\sqrt{58}dx}{7\cos x+3\sin x}$

(where $C$ is constant of integration)

The range of $k$, for which the inequality $k{x}^2-3kx+3<0$, $($where $x\in R)$ holds true is

Match the given angles with the corresponding slopes.

$\begin{array}{cc}Positivex-direction& Slopeofline\\ 1.)0^o& p.)0\\ 2.){90}^o& q.)-\sqrt{3}\\ 3.){120}^o& r.)-\frac{1}{\sqrt{3}}\\ 4.){150}^o& s.)Notdefined\end{array}$

Equation of circle touching the line |x-2|+|y-3|=4 will be

Distance $\left(\text{in units}\right)$ of a point with position vector $\hat{i}+2\hat{j}+\hat{k}$ from the line given by $L:\overrightarrow{r}=2\hat{i}+\hat{j}+3\hat{k}+\lambda (\hat{i}+\hat{j}+\hat{k})$ is:

Let $f\left(x\right)={\int }_1^x\sqrt{2-t^2}\mathrm{dt}$. Then the real roots of the equation $x^2-f\text{'}\left(x\right)=0$ are

If the mean of the set of numbers $x_1,x_2,....x_n$ is $\overline{x}$, then the mean of the numbers $x_i+2i,1\le i\le n$ is

Find the Number of ways such that 5 boys and 5 girls sit together such that boys and girls are in alternate positions

Let R be the relation on the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3,3), (3,2)}. then R is

The number of elements in the sample space for the indicated experiment, where a coin is tossed four times is:

Write
the general term in the expansion of (x²
– yx)¹²,
x ≠
0

$z_1$ and $z_2$ are any two distinct complex numbers in an argand plane. If $\alpha \beta |z_1|=\gamma \delta |z_2|$ , then the complex number $\frac{\alpha \beta z_1}{\gamma \delta z_2}+\frac{\gamma \delta z_2}{\alpha \beta z_1}$ lies on the $(\alpha ,\beta \in R)$

Find the area of the smaller region bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the line $\frac{x}{a}+\frac{y}{b}=1.$

The value of $\underset{n\to \infty }{Lim}\frac{3^{n+1}+2^{n+2}}{{5.3}^n-2^{n-1}}$ is

$\frac{a^{2}sin(B-C)}{sinA}+\frac{b^{2}sin(C-A)}{sinB}+\frac{c^{2}sin(A-B)}{sinC}=0$

Let $(1+x+x^2{)}^{20}(2x+1)=a_0+a_1{x}^1+a_2{x}^2+..........+a_{41}{x}^{41}$ then $\frac{a_0}{1}+\frac{a_1}{2}+\frac{a_2}{3}+............\frac{a_{41}}{42}$ is equal to

If the length of latus rectum of a hyperbola whose eccentricity is $\frac{3}{\sqrt{5}}$, centre $(4,3)$ and axis is parallel to coordinate axis is $8$ units, then the equation(s) of the hyperbola is/are

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $(x_i,y_i)$ for $i=1,2,3,and4$ then $y_1+y_2+y_3+y_4=$

Prove that :

$(1+tan\theta +sec\theta )(1+cot\theta -cosec\theta )$ = 2

${lim}_{x\to 0}\frac{sin5x-sin3x}{sinx}$

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{2}}\frac{co{s}^{2}x}{co{s}^{2}x+4si{n}^{2}x}dx.$

Maximise Z
= 5x + 3y

subject
to.

Two rods whose lengths are $10\text{units}$ and $6\text{units}$ slides along $X$ and $Y$ axes respectively in such a way that their extremities are always concyclic, then the locus of the centre of the circle is

If $A=\left[\begin{array}{cc}5& 3\\ -2& 1\end{array}\right],$ then $3A^2-20A+30I$ is equal to:

The value of definite integral $\underset{0}{\overset{1}{\int }}x(1-x{)}^{n}dx$ is:

1. 50

If $|x|<1$,then the coefficient of $x^n$ in the expansion of $(1+x+x^2+x^3+....{)}^2$ is

Sixteen men compete with one another in running, swimming and riding. How many prize lists could be made if there were altogether $6$ prizes of different values, one for running, $2$ for swimming and $3$ for riding.

If $|x+3|\ge 10$,then

If $p,q$ are the roots of the equation $x^2+px+q=0$ then